Generalizations of the core-EP orthogonality and properties of the core-EP pre-order

[O. M. Stanimirović]

[9], II, 106 str.

Biografija: str. 105. Bibliografija: str. [98-104].

Operator theory

The subject of the scientific research in this doctoral dissertation are various generalizations of the concept of core-EP orthogonality, as well as the properties of the core-EP pre-order. The core-EP inverse was first defined for square matrices of an arbitrary index as a generalization of the core inverse for square matrices of index 1. This concept was subsequently studied for Drazin invertible operators, elements of a ring with involution, and elements of a Banach algebra. The core-EP inverse is applied in solving systems of equations and minimization problems, as well as for defining core-EP orthogonality and the core-EP pre-order. The aim of this dissertation is, primarily, the generalization of the concepts of one-sided core orthogonalities and core-EP orthogonality. One-sided core orthogonalities are introduced for square matrices of index 1, while this dissertation investigates orthogonality concepts for Drazin invertible operators of an arbitrary index. More precisely, left and right core-EP orthogonality and mweak group orthogonality are presented, along with their relations to existing orthogonality concepts. Various characterizations of these new orthogonalities are studied. The significance of these new concepts of orthogonality lies in the study of additivity properties and pre-orders. New properties of the core-EP pre-order are also considered. Specifically, equivalent conditions under which the core-EP pre-order holds are presented, as well as conditions under which the core-EP order implies that the reverse order and forward order laws for the core-EP inverse hold. In a ring with involution, new concepts of orthogonality, which represent generalizations of existing orthogonality concepts for operators and elements of a ring with involution, are explored. Various properties and characteristics of orthogonal elements are proven, and their matrix forms are also presented. In cases where the new generalized orthogonalities hold, the corresponding additive properties are also studied. Furthermore, the properties of the core- EP pre-order and the reverse order and forward order laws for the core-EP inverse are investigated. The results of this doctoral dissertation relate to the generalizations of existing orthogonality concepts to new concepts of orthogonality for operators and elements of a ring with involution. Existing results are generalized and supplemented with new ones.

jezgarni-EP inverz, jezgarna-EP ortogonalnost, delimično uređenje i pre-uređenje

core-EP inverse, core-EP orthogonality, partial orders and pre-order

517.948/.4:512.552(043.3)

P 001; P 120; P 140.

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